<p>
  Then we go back to the stock price. If we use the generalized Wiener process to model the stock price, for example, the share price of a stock is $30 now, then the percentage change in price from now to say the end of this year is normally distributed with constant mean and variance. But there is a problem that for normal distribution assumption of return, there is the possibility that the stock price will be negative this is unrealistic for stock prices. For example, this percentage change could be -1.1, which means the stock price at the end of the year is -$3.
</p>
<p>
  On the other hand, if the spot price is small for a stock, then it tends to have small increments in price over a given time interval, on the contrary, stocks with high prices tend to have much larger increments in price on the same interval. But Wiener process has a variance which depends on just the time interval but not on the price itself. Thus it is not appropriate to assume that a stock price follows a generalized Wiener process with constant drift rate and variance rate.
</p>
<p>
  In order to characterize the dynamics of a stock price process and fix this problem, we model the proportional increase in the stock price by using the stochastic differential equation (SDE):
</p>

\[\text dS_t=\mu S_t\ \text d t+\sigma S_t\ \text dW_t \cdot\cdot\cdot\cdot\cdot\cdot(1)\]
<p>
  Where \(dS_t\) is the change in the stock price over a short time period from \(t\) to \(t+\Delta t\). \(μ\) is the drift term and can be deemed as the annual expected level of the stock return, \(σ\) is the annual volatility of the stock. Here \(dS_t\) at different time \(t\) are independent, which means today's price change is independent with tomorrows change. \(W_t\) is a standard Wiener process (1) we discussed above.
</p>

<p>
  Now in the above equation, the drift and variance rate of stock price \(S\) is not only correlated with time \(t\) but also a function of both \(S\) itself and time \(t\).
</p>

<p>
  The discrete approximation form of (1) is
</p>
\[\Delta S=\mu S\Delta t+\sigma S\epsilon\sqrt{\Delta t}\]
<p>
  We can also derive the process that \(ln(S_t)\) follows (here we just give the result, but the derivation uses the Ito Lemma):
</p>
\[\text d\ ln(S_t)=(\mu-\frac{\sigma^2}{2}) \ \text d t+\sigma \ \text dW_t \cdot\cdot\cdot\cdot\cdot\cdot(2)\]
<p>
  Here \(ln(S_t)\) follows a generalized Wiener process as which has constant drift rate and variance rate. That means the change in \(ln(S_t)\) during time interval \(\Delta t\) is normally distributed. This is the lognormal property of stock prices.
</p>
\[lnS_{t+\Delta t}-lnS_t\sim N\left[(\mu-\frac{\sigma^2}{2})\Delta t,\sigma^2\Delta t\right]\]

\[lnS_{t+\Delta t}\sim N\left[\text ln{S_t}+(\mu-\frac{\sigma^2}{2})\Delta t,\sigma^2\Delta t\right]\]
<p>
  Now the logarithm stock price follows the normal distribution. We write (2) into discrete approximation form as:
</p>
\[ln(S_{t+\Delta t})-ln(S_{t})=(\mu-\frac{\sigma^2}{2})\ \Delta t+\sigma \epsilon\sqrt{\Delta t}\]
<p>Equivalently</p>

\[S_{t+\Delta t}=S_{t}\exp\left[(\mu-\frac{\sigma^2}{2})\ \Delta t+\sigma \epsilon\sqrt{\Delta t}\right]\cdot\cdot\cdot\cdot\cdot\cdot(3)\]
<p>
  If we change t to 0 and change \(\Delta t\) to T, we get
</p>
\[S_{T}=S_{0}\exp\left[(\mu-\frac{\sigma^2}{2})\ T+\sigma \epsilon\sqrt{T}\right]\]
<p>
  According to the above equation, we know the stock price at time \(T\) should always be greater than 0 and the problem of having a negative price is fixed. Thus we say the stock price \(S\) is a Geometric Brownian motion because the logarithm of \(S\) follows a Brownian motion.
</p>
